Editing Bell polynomials (section)

==Combinatorial meaning==

The exponential Bell polynomial encodes the information related to the ways a set can be partitioned. For example, if we consider a set {A, B, C}, it can be partitioned into two non-empty, non-overlapping subsets, which are also referred to as parts or blocks, in 3 different ways:

:{{A}, {B, C}}
:{{B}, {A, C}}
:{{C}, {B, A}}

Thus, we can encode the information regarding these partitions as

:<math>B_{3,2}(x_1,x_2) = 3 x_1 x_2. </math>

Here, the subscripts of ''B''<sub>3,2</sub> tell us that we are considering the partitioning of a set with 3 elements into 2 blocks. The subscript of each ''x''<sub>i</sub> indicates the presence of a block with ''i'' elements (or block of size ''i'') in a given partition. So here, ''x''<sub>2</sub> indicates the presence of a block with two elements. Similarly, ''x''<sub>1</sub> indicates the presence of a block with a single element. The exponent of ''x''<sub>i</sub><sup>j</sup> indicates that there are ''j'' such blocks of size ''i'' in a single partition. Here, the fact that both ''x''<sub>1</sub> and ''x''<sub>2</sub> have exponent 1 indicates that there is only one such block in a given partition. The coefficient of the [[monomial]] indicates how many such partitions there are. Here, there are 3 partitions of a set with 3 elements into 2 blocks, where in each partition the elements are divided into two blocks of sizes 1 and 2.

Since any set can be divided into a single block in only one way, the above interpretation would mean that ''B''<sub>''n'',1</sub> = ''x''<sub>''n''</sub>. Similarly, since there is only one way that a set with ''n'' elements be divided into ''n'' singletons, ''B''<sub>''n'',''n''</sub> = ''x''<sub>1</sub><sup>''n''</sup>.

As a more complicated example, consider

:<math>B_{6,2}(x_1,x_2,x_3,x_4,x_5)=6x_5x_1+15x_4x_2+10x_3^2.</math>

This tells us that if a set with 6 elements is divided into 2 blocks, then we can have 6 partitions with blocks of size 1 and 5, 15 partitions with blocks of size 4 and 2, and 10 partitions with 2 blocks of size 3.

The sum of the subscripts in a monomial is equal to the total number of elements. Thus, the number of monomials that appear in the partial Bell polynomial is equal to the number of ways the integer ''n'' can be expressed as a summation of ''k'' positive integers. This is the same as the [[integer partition]] of ''n'' into ''k'' parts. For instance, in the above examples, the integer 3 can be partitioned into two parts as 2+1 only. Thus, there is only one monomial in ''B''<sub>3,2</sub>. However, the integer 6 can be partitioned into two parts as 5+1, 4+2, and 3+3. Thus, there are three monomials in ''B''<sub>6,2</sub>.  Indeed, the subscripts of the variables in a monomial are the same as those given by the integer partition, indicating the sizes of the different blocks. The total number of monomials appearing in a complete Bell polynomial ''B<sub>n</sub>'' is thus equal to the total number of integer partitions of&nbsp;''n''.

Also the degree of each monomial, which is the sum of the exponents of each variable in the monomial, is equal to the number of blocks the set is divided into. That is, ''j''<sub>1</sub> + ''j''<sub>2</sub> + ... = ''k'' . Thus, given a complete Bell polynomial ''B<sub>n</sub>'', we can separate the partial Bell polynomial ''B<sub>n,k</sub>'' by collecting all those monomials with degree ''k''.

Finally, if we disregard the sizes of the blocks and put all ''x''<sub>''i''</sub> = ''x'', then the summation of the coefficients of the partial Bell polynomial ''B''<sub>''n'',''k''</sub> will give the total number of ways that a set with ''n'' elements can be partitioned into ''k'' blocks, which is the same as the [[Stirling numbers of the second kind]]. Also, the summation of all the coefficients of the complete Bell polynomial ''B<sub>n</sub>'' will give us the total number of ways a set with ''n'' elements can be partitioned into non-overlapping subsets, which is the same as the Bell number.

In general, if the integer ''n'' is [[integer partition|partitioned]] into a sum in which "1" appears ''j''<sub>1</sub> times, "2" appears ''j''<sub>2</sub> times, and so on, then the number of [[partition of a set|partitions of a set]] of size ''n'' that collapse to that partition of the integer ''n'' when the members of the set become indistinguishable is the corresponding coefficient in the polynomial.

===Examples===

For example, we have

:<math>B_{6,2}(x_1,x_2,x_3,x_4,x_5)=6x_5x_1+15x_4x_2+10x_3^2</math>

because the ways to partition a set of 6 elements as 2 blocks are

:6 ways to partition a set of 6 as 5&nbsp;+&nbsp;1,
:15 ways to partition a set of 6 as 4&nbsp;+&nbsp;2, and
:10 ways to partition a set of 6 as 3&nbsp;+&nbsp;3.

Similarly,

:<math>B_{6,3}(x_1,x_2,x_3,x_4)=15x_4x_1^2+60x_3x_2x_1+15x_2^3</math>

because the ways to partition a set of 6 elements as 3 blocks are

:15 ways to partition a set of 6 as 4&nbsp;+&nbsp;1&nbsp;+&nbsp;1,
:60 ways to partition a set of 6 as 3&nbsp;+&nbsp;2&nbsp;+&nbsp;1, and
:15 ways to partition a set of 6 as 2&nbsp;+&nbsp;2&nbsp;+&nbsp;2.